Types of Functions

1. One-One Function (Injective)

A function is called one-one if different elements of domain map to different elements of codomain. No two inputs share the same output.

• f(a) = f(b) ⇒ a = b
• Each output is used only once

Example: f(x) = x + 1

2. Many-One Function

A function where two or more inputs map to the same output. This is allowed in functions.

• Different inputs → same output

Example: f(x) = x²

3. Onto Function (Surjective)

A function is onto if every element of codomain has at least one pre-image in domain.

• Range = Codomain
• No element in codomain is left unused

Example: f(x) = x³ (R → R)

4. Into Function

A function is into if at least one element of codomain has no pre-image.

• Range ≠ Codomain
• Some outputs are not used

Example: f(x) = x² (R → R)

Quick Comparison

• One-One → unique outputs
• Many-One → same output allowed
• Onto → all outputs used
• Into → some outputs unused